Theorem nfcsb1d 2880

Description: Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)

Hypothesis

Ref	Expression
nfcsb1d.1	|- ( ph -> F/_ x A )

Assertion

Ref	Expression
nfcsb1d	|- ( ph -> F/_ x [_ A / x ]_ B )

Proof of Theorem nfcsb1d

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-csb 2853	. 2 |- [_ A / x ]_ B = { y | [. A / x ]. y e. B }
2		nfv 1421	. . 3 |- F/ y ph
3		nfcsb1d.1	. . . 4 |- ( ph -> F/_ x A )
4	3	nfsbc1d 2780	. . 3 |- ( ph -> F/ x [. A / x ]. y e. B )
5	2, 4	nfabd 2196	. 2 |- ( ph -> F/_ x { y | [. A / x ]. y e. B } )
6	1, 5	nfcxfrd 2176	1 |- ( ph -> F/_ x [_ A / x ]_ B )

Syntax hints:   -> wi 4   e. wcel 1393   {cab 2026   F/_wnfc 2165   [.wsbc 2764   [_csb 2852

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-sbc 2765  df-csb 2853

This theorem is referenced by:  nfcsb1  2881